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is obscured; when the Reasonings of his Science seem anywhere to oppose each other, or become in any part too complex or too little valid for his belief to rest firmly upon them; or when, though trial may have taught him that a rule is useful, or that a formula gives true results, he cannot prove that rule, nor understand that formula: when he cannot rise to intuition from induction, or cannot look beyond the signs to the things signified. “It is not here asserted that every or any Algebraist belongs exclusively to any one of these three schools, so as to be only Practical, or only Philological, or only Theoretical. Language and Thought react, and Theory and Practice help each other. No man can be so merely practical as to use frequently the rules of Algebra, and never to admire the beauty of the language which expresses those rules, nor care to know the reasoning which deduces them. No man can be so merely philological an Algebraist but that things or thoughts will at some times intrude upon signs; and occupied as he may habitually be with the logical building up of his expressions, he will feel sometimes a desire to know what they mean, or to apply them. And no man can be so merely Theoretical, or so exclusively devoted to thoughts, and to the contemplation of theorems in Algebra, as not to feel an interest in its notation and language, its symmetrical system of signs, and the logical forms of their combinations; or not prize those practical aids, and especially those methods of research, which the discoveries and contemplations of Algebra have given to other sciences. But, distinguishing without dividing, it is perhaps correct to say that every Algebraical Student and every Algebraical Composition may be referred upon the whole to one or other of these three schools, according as one or other of these three views habitually actuates the man, and eminently marks the work. “These remarks have been premised, that the reader may more easily and distinctly perceive what the design of the following communication is, and what the Author hopes or at least desires to accomplish. That design is Theoretical, in the sense already explained, as distinguished from what is Practical on the one hand, and from what is Philological on the other. The thing aimed at, is to improve the Science, not the Art, nor the Language of Algebra. The imperfections sought to be removed, are confusions of thought, and obscurities or errors of reasoning; not difficulties of application of an instrument, nor failures of symmetry in expression. And that confusions of thought and errors of reasoning, still darken the beginnings of Algebra, is the earnest and just complaint of sober and thoughtful men, who in a spirit of love and honour have studied Algebraic Science, admiring, extending, and applying what has been already brought to #. and feeling all the beauty and consistence of many a remote deduction, from principles which yet remain obscure, and doubtful. “For it has not fared with the principles of Algebra as with the principles of Geometry. No candid and intelligent person can doubt the truth of the chief properties of Parallel Lines,
as set forth by Euclid in his Elements, two thousand years ago; though he may well desire to see them treated in a clearer and better method. The doctrine involves no obscurity nor confusion of thought, and leaves in the mind no reasonable ground for doubt, although ingenuity may usefully be exercised in inproving the plan of the argument. But it requires no peculiar scepticism to doubt, or even to disbelieve, the doctrine of Negatives and Imaginaries, when set forth (as it has commonly been) with principles like these: that a greater magnitude may be subtracted from a less, and that the remainder is less than nothing ; that two negative numbers, or numbers denoting magnitudes, each less than nothing, may be multiplied the one by the other, and that the product will be a positive number, or a number denoting a magnitude greater than nothing; and that although the square of a number, or the product obtained by multiplying that number by itself, is therefore always positive, whether the number be positive or negative, yet that numbers, called imaginary, can be found or conceived or determined, and operated on by all the rules of positive and negative numbers, as if they were subject to those rules, although they have negatire squares, and must therefore be supposed to be themselves neither positive nor negative, nor yet null numbers, so that the magnitudes which they are supposed to denote can neither be greater than nothing, nor less than nothing, nor even equal to nothing. It must be hard to found a SCIENCE on such grounds as these, though the forms of logic may build up from them a symmetrical system of expressions, and a practical art may be learned of rightly applying useful rules which seem to depend upon them. “So useful are those rules, so symmetrical those expressions, and yet so unsatisfactory those principles from which they are supposed to be derived, that a growing tendency may be perceived to the rejection of that view which regarded Algebra as a SciENCE, in some sense analogous to Geometry, and to the adoption of one or other of those two different views, which regard Algebra as an Art, or as a Language: as a System of Rules, or else as a System of Expressions, but not as a System of Truths, or Results having any other validity than what they may derive from their practical usefulness, or their logical or philological coherence. Opinions thus are tending to substitute for the Theoretical question,- Is a Theorem of Algebra true?” the Practical question,-‘Can it be applied as an Instrument, to do or to discover something else, in some research which is not Algebraical ?” or else the Philological question,-‘Does its exession harmonize, according to the Laws of }. uage, with other Algebraical expressions o' “Yet a natural regret might be felt, if such were the destiny of Algebra; if a study, which is continually engaging mathematicians more and more, and has almost superseded the Study of Geometrical Science, were found at last to be not, in any strict and proper sense, the Study of a Science at all: and if, in thus exchanging the ancient for the modern Mathesis, there were a gain only of Skill or Elegance, at the expense of Contemplation and Intuition. Indulgence, therefore, may be hoped for, by any one who would inquire, whether existing Algebra, in the state to which it has been already unfolded by the masters of its rules and of its language, offers indeed no rudiment which may encourage a hope of developing a SCIENCE of Algebra: a Science properly so called; strict, pure, and independent; deduced by valid reasonings from its own intuitive principles; and thus not less an object of priori contemplation than Geometry, nor less distinct, in its own essence, from the Rules which it may teach or use, and from the Signs by which it may express its meaning. “The Author of this paper has been led to the belief, that the Intuition of Time is such a rudiment. This belief involves the three following as components: First, that the notion of Time is connected with existing Algebra; Second, that this notion or intuition of Time may be unfolded into an independent Pure Science; and Third, that the science of Pure Time, thus unfolded, is co-extensive and identical with Algebra, so far as Algebra itself is a Science. The first component judgment is the result of an induction; the second of a deduction; the third is a joint result of the deductive and inductive processes.”
It would not be easy, in our limited space, and without using algebraic symbols freely, to give the reader more than a very vague idea of the nature of this Essay. What we are most concerned with at present is the bearing of its processes upon the interpretation of imaginary quantities, and even on that we can only say a few words. The step in time from one definite moment to another depends, as is easily seen, solely on the relative position in time of the two moments, not on the absolute date of either. And, in &omparing one such step with another, there can be difference only in duration and direction,-i.e., one step may be longer or shorter than the other, and the two may be in the same or in opposite directions, progressive or retrograde. Here numerical factors, positive and negative, come in. But to introduce something analogous to the imaginary of algebra, Hamilton had to compare with each other, not two, but two pairs or Couples of, steps. Thus, a and b representing steps in time, (a, b) is called a Couple; and its value depends on the order as well as the magnitude of its constituent steps. It is shown that (–a,—b) is the same couple taken negatively. And the imaginary of common algebra is now represented by that operation on a step-couple which changes the sign (or order of progression) of the second step of the couple, and makes the steps change places. That is, it is the factor or operator which changes (a, b) into (–b,a): for a second application will obviously produce the result (-a-b). There is, no doubt, here a perfectly
real interpretation for the so-called imaginary quantity, but it cannot be called simple, nor is it at all adapted for elementary instruction. The reader will observe that Hamilton, with his characteristic sagacity, has chosen a form of interpretation which admits of no indeterminateness. Unlike Wallis and others, who strove to express ordinary algebraic imaginaries by directions in space, Hamilton gave his illustration by time or progression, which admits, so to speak, of but one dimension. We may attempt to give a rough explanation of his process, for the reader who is not familiar with algebraic signs, in some such way as this:—If an officer and a private be set upon by thieves, and both be plundered of all they have, this operation may be represented by negative unity. And the imaginary quantity of algebra, or the square root of negative unity, .# then be represented by a process which would rob the private only, but at the same time exchange the ranks of the two soldiers. It is obvious that on a repetition of this process both would be robbed, while they would each be left with the same rank as at first. But what is most essential for remark here is that the operation corresponding to the so-called imaginary of algebra is throughout regarded as perfectly real.
In 1835 Hamilton seems to have extended the above theory from Couples to Triplets, and even to a general theory of Sets, each containing an assigned number of time-steps. Many of #. results are extremely remarkable, as may be gathered from the only published account of them, a brief notice in the Preface to his Lectures on Quaternions. After having alluded to them, he proceeds: “There was, however, a motive which induced me then to attach a special importance to the consideration of triplets. . . . This was the desire to connect, in some new and useful (or at least interesting) way, calculation with geometry, through some undiscovered extension, to space of three dimensions, of a method of construction or representation which had been employed with success by Mr. Warren (and indeed also by other authors, of whose writings I had not then heard), for operations on right lines in one plane : which method had given a species of geometrical interpretation to the usual and well-known imaginary symbol of algebra.” After many attempts, most of which launched him, like his predecessors and contemporaries, into a maze of expressions of fearful complexity, he suddenly lit upon a system of extreme simplicity and elegance. The following remarkably interesting extract from a letter gives his own account of the discovery:
“ 2 Oct. 15, '58.
“P. S.–To-morrow will be the fifteenth birthday of the Quaternions. They started into life, or light, full grown, on D the 16th of October 1843, as I was walking with Lady Hamilton to Dublin, and came up to Brougham Bridge, which my boys have since called the Quatermion Bridge. That is to say, I then and there felt the galvanic circuit of thought close; and the sparks which fell from it were the fundamental equations between i, j, k, eractly such as I have used them ever since. I pulled out, on the spot, a pocket-book which still exists, and made an entry, on which, at the very moment, I felt that it might be worth my while to expend the labour of at least ten (or it might be fifteen) years to come. But then, it is fair to say that this was because I felt a problem to have been at that moment solved,—an intellectual want relieved,—which had haunted me for at least fifteen years before.
“Less than an hour elasped, before I had asked and obtained leave, of the Council of the Royal Irish Academy, of which Society I was, at that time, the President, to read at the merz general Meeting, a Paper on Quaternions; which I accordingly did, on November 13, 1843.
“Some of those early communications of mine to the Academy may still have some interest for a person like you, who has since so well studied my Volume, which was not published for ten years afterwards.
“In the meantime, will you not do honour to the birthday, to-morrow, in an extra cup of —ink? for it may be obsolete now to propose XXX,+or even XYZ.”
We must now endeavour to explain, in as popular a manner as possible, the nature of the new Calculus. In order to do so, let us recur to the suggestion of Wallis, before described, and endeavour to ascertain the exact nature of its defects. We easily see that one great defect is want of symmetry. As before stated, if we take an eastward line, of proper length, to represent positive unity, an equal westward line represents negative unity; but all lines perpendicular or inclined to these are represented by socalled imaginary quantities. Hamilton's great step was the attainment of the desired symmetry by making all lines alike expressible by so-called imaginary quantities. Thus, instead of 1 for the eastward line, and the square root of negative unity for a northward line, he represents every line in space whose length is unity by a distinct square root of negative unity. All are thus equally imaginary, or rather equally real. The i,j, k mentioned in the extract just given, are three such quantities; which (merely for illustration, because any other set of three mutually rectangular directions will do as well) we may take as representing unit lines drawn respectively eastwards, northwards, and upwards. The square of each being
negative unity, we may interpret the effect
of such a line (when used as a factor or operator) as a left (or right) handed rotation through one right angle about its direction. The effect of a repetition of the operation is a rotation through two right angles, or a simple inversion. Thus, if we operate with i on f we turn the northward line left-handedly through a right angle about an eastward axis, i.e., we raise it to a direction vertically upward. Thus we see that j=k. But, if we perform the operation again, we see that ik or is is now the southward line or -j. Thus "E-1. And similarly with the squares of j and k. It is to be noticed that in all these cases the operating line is supposed to be perpendicular to the operand. Also that we have taken for granted (what is easily proved), that i, j, k may stand indifferently for the unit line themselves or for the operation of turning through a right angle. Thus, the equation is =k may either mean (as above) that i acting on the line j turns it into the line k; or that the rectangular rotation f, succeeded by i, is equivalent to the single rotation k. We may easily verify the last assertion by taking i as the operand. j changes it to -k, and i changes this to j. But k turns i into 7 at once.
Even these ... ideas lead us at once to one of the most remarkable properties of quarternions. When turning the northward line (j) about the eastward line (i), we wrote the operator first-thus ij=k. Now, the order of multiplication is not indifferent, for ji is not equal to k. ji, in fact, expresses that i (the eastward line) has been made to rotate left-handedly through a right angle about j (the northward line). This obviously brings it to the downward direction, or we have ji= -k, Similar expressions hold for the other products two and two of the three symbols. Thus we have the laws of their multiplication complete. And on this basis the whole theory may be erected. Now any line whatever may be resolved (as velocities and forces are) into so much eastward (or westward), so much northward (or southward), and so much upward (or downward). Hence every line may be expressed as the sum of three parts, numerical multiples of i, j, and k respectively. Call these numbers 2, y, and 2, then the line may be expressed by zi+yj + zk. If we square this we find -(z” +y^+2*); for the other terms occur in phirs like zi xuj and yj xzi, and destroy each other; since we have, as above, ij-i-ji=0, with similar results for the other pairs of the three rectangular unit-lines. Now z*-ī-y?--z" is the square of the length of the line (by a double application of Euclid I. xlvii.) Hence the square of every
line of unit length is negative unity. And herein consists the grand symmetry and consequent simplicity of the method; for we may now write a single symbol such as o. (Greek letters are usually employed by Hamilton in this sense), instead of the much more cumbrous and not more expressive form zi-Hyj + zk. We have seen that the product, and consequently the quotient, of two lines at right angles to each other, is a third line perpendicular to both; and that the product or quotient of two parallel lines is a number: what is the product, or the quotient, of two lines not at right angles, and not parallel, to each other ? It is a QUATERNION. This is very easily seen thus. Take the case of the quotient of one line by another, and suppose them drawn from the same point: the first line may, by letting fall a perpendicular from its extremity upon the second, be resolved into two parts, one parallel, the other perpendicular, to the second line. The quotient of the two parallel lines is a mere number, that of the two perpendicular lines is a line, and can therefore be expressed as the sum of multiples of i, j, and k. Hence, w representing the numeral quotient of the parallel portions, the quotient of any two lines, may be written as w-Hzi+yj +zk; and, in this form, is seen to depend essentially upon Four perfectly distinct numbers; whence its name. In actually working with quaternions, however, this cumbrous form is not necessary; we may express it as B-i-a, B and a being the two lines of which it is the quotient; and in various other equivalent algebraic forms; or we may at once substitute for it a single letter (Hamilton uses the early letters, a, b, c, etc.) The amount of condensation, and consequent shortening, of the work of any particular problem which is thus attained, though of immense importance, is not by any means the only or even the greatest advantage possessed by quaternions over other methods of treating analytical geometry. They render us entirely independent of special lines, axes of co-ordinates, etc., devised for the application of other methods, and take their reference lines, in every case from the particular Fo to which they are applied. They ave thus what Hamilton calls an “internal” character of their own; and give us, without trouble, an insight into each special question, which other methods only yield to a combination of great acuteness with patient labour. In fact, before their invention, no process was known for treating problems in tridimensional space in a thoroughly natural and inartificial manner. But let him speak for himself. The fol
lowing passage is extracted from a letter to a mathematical friend, who was at the time engaged in studying the new calculus:—
- “Whatever may be the future success of Quaternions as an Instrument of Investigation, they furnish already, to those who have learned to read them, (povávra aruverotorw,) a powerful ORGAN of ExPREssion, especially in geometrical science, and in all that widening field of physical inquiry, to which relations of space (not always easy to express with clearness by the Cartesian Method) are subsidiary, or rather are indispensable.”
To follow up the illustration we began with, it may be well merely to mention here that a quaternion may in all cases be represented as a power of a line. When a line is raised to the first, third, or any odd integral power, it represents a right-angled quaternion, or one which contains no pure numerical part; when to the second, fourth, or other even integral power, it degenerates into a mere number; for all other powers it contains four distinct terms. Compare this with the illustration already given, leading to De Moivre's theorem; and we see what a grand step Hamilton supplied by assigning in every case a definite direction to the axis about which the rotation takes place.
There are many other ways in which we can exhibit the essential dependence of the product or quotient of two directed lines (or Vectors as Hamilton calls them), on four numbers (or Scalars), and the consequent fitness of the name Quaternion. It may interest the reader to see another of them. Let us now regard the quaternion as the factor or operator required to change one side of a triangle into another; and let us suppose the process to be performed by turning one of the sides round till it coincides in direction with the other, and then stretching or shortening it till they coincide in length also. For the first operation we must know the axis about which the rotation is to take place, and the angle or amount of rotation. Now the direction of the axis depends on two numbers (in Astronomy they may be Altitude and Azimuth, Right Ascension and Declination, or Latitude and Longitude); the amount of rotation is a third number; and the amount of stretching or shortening in the final operation is the fourth.
Among the many curious results of the invention of quaternions, must be noticed the revival of fluzions, or, at all events, a mode of treating differentials closely allied to that originally introduced by Newton. The really useful, but over-praised differential coefficients, have, as a rule, no meaning in quaternions; so that, except when dealing with scalar variables (which are simply degraded quaternions), we must employ in differentiation fluxions or differentials. And the reader may easily understand the cause of this. It lies in the fact that quaternion multiplication is not commutative; so that, in differentiating a product, for instance, each factor must be differentiated where it stands; and thus the differential of such a product is not generally a mere algebraic multiple of the differential of the independent quaternion-variable. It is thus that the whirligig of time brings its revenges. The shameless theft which Leibnitz committed, and which he sought to disguise by altering the appearance of the stolen goods, must soon be obvious, even to his warmest partisans. They can no longer pretend to regard Leibnitz as even a second inventor when they find that his only possible claim, that of devising an improvement in notation, merely unfits Newton's method of fluxions for application to the simple and symmetrical, yet massive, space-geometry of Hamilton. One very remarkable speculation of Hamilton's is that in which he deduces, by a species of metaphysical or a priori reasoning, the results previously mentioned, viz., that the product (or quotient) of two parallel vectors must be a number, and that of two mutually perpendicular vectors a third perpendicular to both. We cannot give his reasoning at full length, but will try to make part of it easily intelligible. Suppose that there is no direction in space pre-eminent, and that the product of two vectors is something which has quantity, so as to vary in amount if the factors are changed, and to have its sign changed if that of one of them is reversed; if the vectors be parallel, their product cannot be, in whole or in part, a vector inclined to them, for there is nothing to determine the direction in which it must lie. It cannot be a vector parallel to them; for by changing the sign of both factors the product is unchanged, whereas, as the whole system has been reversed, the product vector ought to have been reversed. Hence it must be a number. Again, the product of two perpendicular vectors cannot be wholly or partly a number, because on inverting one of them the sign of that number ought to change; but inverting one of them is simply equivalent to a rotation through two right angles about the other, and from the symmetry of space ought to leave the number unchanged. Hence the product of two perpendicular vectors must be a vector, and an easy extension of the same reasoning shows that it WOL. XLV.
as it is called in the Carmen Aureum. Of course, so far as mere derivation goes, it is hard to see any difference between the Tetractys and the Quaternion. But we are almost entirely ignorant of the meaning Pythagoras attached to his mystic idea, and it certainly must have been excessively vague, if not quite so senseless as the Abracadabra of later times. Yet there is no doubt that Hamilton was convinced that Quaternions, in virtue of some process analogous to the quasi-metaphysical speculation we have just sketched, are calculated to lead to important discoveries in physical science; and, in fact, he writes—
“Little as I have pursued such [physical] Studies, even in books, you may judge from my Presidential Addresses, pronounced on the occasions of delivering Medals (long ago), from the chair of the R.I.A., to Apjohn and to Kane, that physical (as distinguished from mathematical) investigations have not been wholly alien to my somewhat wide, but doubtless very superficial, course of reading. You might, with
" out offence to me, consider that I abused the
license of hope, which may be indulged to an inventor, if I were to confess that I expect the Quaternions to supply hereafter, not merely mathematical methods, but also physical suggestions. And, in Fo you are quite welcome to smile, if I say that it does not seem extravagant to me to suppose that a full possession of those d priori principles of mine, about the multiplication of vectors—including the Law of the Four Scales, and the Conception of the Extra-spatial Unit, which have as yet not been much more than hinted to the public—Might have led (I do not at all mean that in my hands they ever would have done so,) to an ANTICIPATION of something like the grand discovery of OERSTED: who, by the way, was a very a priori (and poetical) sort of man himself, as I know from having conversed with him, and received from him some printed pamphlets, several years ago. It is impossible to estimate the chances given, or opened up, by any new way of looking at things; especially when that way admits of being intimately combined with calculation of a most rigorous kind.”
This idea is still further developed in the following sonnet, which gives besides a good